What is the surface area produced by rotating f(x)=x^3-8, x in [0,2]f(x)=x3−8,x∈[0,2] around the x-axis?

1 Answer
Skewd
Jun 21, 2018

first find dS, note the radius of the rotation is x.
dS=sqrt(1+9x^4)dxdS=√1+9x4dx
int_0^2 2pixdS∫202πxdS
to solve make a substitution of w=3x^2w=3x2

Explanation:

If you make the substitution with w your integral becomes
int_0^12 pi/3*sqrt(1+w^2)dw∫120π3⋅√1+w2dw
with a trigonometric substitution w=tan(theta)w=tan(θ)
the integral becomes
pi/3 int_0^arctan(12) sec^3(theta)d(theta)π3∫arctan(12)0sec3(θ)d(θ)
which you can solve by parts or using a table.

Related questions
See all questions in Determining the Surface Area of a Solid of Revolution
Impact of this question
2088 views around the world
You can reuse this answer
Creative Commons License